Imperfect competition, joint harm and market share liability
Maxime CHARREIREy and Eric LANGLAISz
December 29, 2016
Abstract
In this paper, we analyze the output and care decisions of …rms competiting à la Cournot, when
they manufacture a good that may jointly harm some victims (di¤erent from their customers). We
consider di¤erent liability sharing arrangements (per capita vs market share) to which …rms are
exposed, when the expected harm is related more than proportionally to the output of the industry.
We …nd that compared to the per capita rule, the market share apportionment is better for victims
and worse for consumers, in the sense that it leads to a lower level of output and higher expenditures
in care. However, the net e¤ect on the expected harm to victims is ambiguous, such that it is not
clear that the market share rule is dominating in contexts where there is a hard uncertainty, such
that it is not feasible to disentangle the in‡uence of each …rm on the harm borne by victims. We
also show that no sharing arrangement induce the optimal levels of output and care expenditures.
On the other hand, the analysis of the determinants of optimal decisions (output and care) and their
comparison to the equilibrium levels appear as quite deceptive, in the sense that there is no reason to
believe that the optimal level of output (care expenditures) is smaller (larger) than the equilibrium
one attained under imperfect competition.
We are thankful to participants at conferences AFED Paris 2016, ISLE Torino 2016, for their comments and suggestions.
UMR 7235 CNRS and Paris Ouest Nanterre, 200 Avenue de la République, 92001 Nanterre cedex, France.
E-mail: [email protected]
z EconomiX UMR 7235 CNRS and Paris Ouest Nanterre, 200 Avenue de la République, 92001 Nanterre cedex, France.
E-mail: [email protected]
y EconomiX
1
1
Introduction
Following the recent Distilbène litigation 1 , one have acknowledged a revival of the debate in France in
tort cases with joint and several liability. Some French scholars (Molfessis 2015, Quézel-Ambrunaz 2010)
have argued in favor of traditional solutions adopted for damages apportionment (leading to an equal
allocation of the damage) compared to the market share rule from the perspective of consistency to
admitted theories of causation, and adequacy to jurisprudence. Others have motivated the latter rule on
the grounds that market shares may be a proxy for the likelihood of individual liability, in contexts of
joint liability characterized by hard uncertainty and ambiguous causation (Ferey and G’sell 2013, G’sell
2010).2
The market shares rule in tort law …rst appeared in 1980 in the Californian case Sindell v. Abbott
Laboratories, and up to now, Courts in USA have limited its use in the area of product liability in cases
where the origin of the product is unknown/cannot be determined although the set of potential o¤enders
is perfectly identi…ed. However, the potential for the extension of the market share liability is broader
both in scope (domains of law) and space. A typical example is illustrated by the progressive di¤usion
of environmental liability all around the world. At the European level, the Directive 2004/35/CE on
environmental liability aims at "inducing operators to adopt measures and develop practices to minimize
the risks of environmental damage". The directive let to the member states law the determination of the
liability apportionment in case of multi party causation. Interestingly, the french law, at the article L.
162-18 of the environmental code establishes that when an environmental harm has multiple causation,
then the damage must be divided among operators in proportion of their participation to the harm3 .
Although Courts have used the traditional solutions (including dividing the damages to victims equally
between o¤enders) up to now, we believe that exists a potential for a shift to the solution of the market
share rule. The main motivation is that situations having the potential to harm environment in relation
with industrial and human activities (point-source pollutions) are characterized by hard uncertainty: it
may be di¢ cult, even impossible, to disentangle the in‡uence of each individual o¤ender on the total
harm to victims.
Indeed, a large range of industrial activities, from the chemical to the oil sectors, have the power to
provoke joint harm to several victims as third parties, or more broadly speaking to the environment. From
this perspective, one also have to acknowledge that in raising the issue of liability and liability sharing,
it turns out to be di¢ cult to ignore the market structure where …rms operate, and the imperfection
of competition. Indeed, a crucial aspect regarding the extension of liability sharing arrangements to
1 See TGI Nanterre 10 april 2014 n 12/12349 and n 12/13064. Interestingly, both cases concerns the diethylstilbestrol
(DES) , a product delivered to pregnant women and which caused years later injuries to the children exposed in utero.
2 See also Dillbary (2011) for a di¤erent perspective on this debate.
3 Article L. 162-18 du Code de l’environnement: Lorsqu’un dommage à l’environnement a plusieurs causes, le coût des
mesures de prévention ou de réparation est réparti par l’autorité visée au 2 de l’article L. 165-2 entre les exploitants, à
concurrence de la participation de leur activité au dommage ou à la menace imminente de dommage.
2
industrial activities relates to their impacts on the strategic interactions between …rms, and the intensity
of competition. This the central issues of our paper.
Regarding the debates discussed above (liability sharing rules and internal consistency of law, or as
proxy of probabilities), we take a agnostic view, and rather focus on a related point although neglected,
which is the impact of liability sharing on market under imperfect competition, and the implications for
the society/economy. Important is to remind that we consider here the potential of extension outside
of product liability, but cases such as environmental law, competition law etc. The questions we ask, in
cases absent of any (knowledge of the) possibility to disentangle multiple o¤enders’ responsibility, are:
which liability sharing arrangement (no liability, equal share, market share) can be considered as the best
outcome ? Does it allow to reach an e¢ cient outcome?
To do so, we use the basic framework of a duopoly in quantity à la Cournot, analyzing the simultaneous
choice of care and output by two competitors producing an homogenous good. The expected harm to third
parties who are not consumers of the good, is related to the market supply of good and to the aggregate
expenditures in precaution. Firms operate in the industry under a rule of strict liability, augmented of
a rule of damages apportionment. On the one hand, the harm can be equally shared among the …rms
(per capita rule), which is one of most common rule used by the Courts. On the other hand, the harm
can shared among …rms in proportion of their market share (market share liability). We believe that two
other features of our set up are also speci…c to the problematic of environmental liability: …rst, we assume
that the level of activity in the industry a¤ects the expected harm to victims in a cumulative way, i.e. the
expected harm per unit of output increases with the total industry output at a more-than-proportional
rate.4 ; second, we assume that precaution is "durable" in the sense of Nussim and Tabbach (2009), that
precautionary measures "may be e¤ective or endure for all activity level, and certainly need not to be
taken per unit of activity". As a typical example of durable precaution, one may think to investment in
speci…c infrastructures corresponding to a (large) …xed cost, independent from the level of output.
In this set-up, two important results emerge. The …rst one is that, regarding the objective of safety
(preserving victims wellbeing), no liability sharing regime strictly dominates the other one: the equilibrium output level of the industry is larger under a per capita rule than under a market share apportionment – which makes victims worse o¤ ex post, since the harm in case of accident is higher. In contrast,
and related to this …rst e¤ect on the output, the equilibrium level of care expenditures is lower under
the market share liability than under the equal sharing arrangement – which deteriorates the situation
ex ante of the victims. The second result is that no sharing arrangement has the power to mimic the
optimal levels of output and care expenditures. Maybe more important, the analysis of the determinants
and comparison of optimal decisions to equilibrium levels of output and care display some uncomfortable, and clearly deceptive results: generally speaking, there is no necessity for the optimal output (care
4 See also Daughety and Reinganum (2014) for a discussion in the context of product liability in the domain of medecine,
food safety, but also pollution.
3
expenditures) to be smaller (larger) than what emerges market discipline and the incentives create by
liability in the context of imperfect competition.
Section 2 present a brief survey of the literature. Section 3 introduces the model, and the benchmark
associated with optimal decisions. Section 4 analyzes the equilibrium under Cournot competition under
two di¤erent liability sharing arrangements, the per capita vs market sharing rule. Section 5 compares
the two equilibrium outcomes with liability sharing to the social optimum. Section 6 concludes.
2
Relation to the literature
The …rst literature to which our paper is connected is about joint and several liability in contexts with
multiple tortfeasors (see Kornhauser and Revesz (2000) for a review). Focusing on the incentives to take
care, Landes and Posner (1980) and Shavell (1985,1987) have …rst considered the standard question of
the comparative advantages of strict liability vs negligence. Later on, Kornhauser and Revesz (1989) and
Miceli and Segerson (1991) turn to the issue …rms entry, but do not consider explicit strategic interactions
between …rms.
Also clearly related to our work is the literature about product liability and imperfect competition.
The seminal papers by Polinsky (1980), Polinsky and Rogerson (1983) who …rst discussed the interplay
between market power and standard liability rules have been extended to di¤erent market set up in
the recent period (Baniak and Grajzl 2016, Baumann and Friehe 2015; Baumann and Friehe and Rasch
2016; Chen and Hua 2015; Daughety and Reinganum 2014). Contrary to the issues discussed here, those
works are not considering the situation of third parties as victims, (i.e. victims without any contractual
arrangement or market relationship with the industry), nor the context of joint liability, or the comparison
of alternative liability sharing arrangements. Also worthy to note is the paper by Hamilton and Sunding
(2000) who discuss the issue of …rms entry in an asymmetric (quantity) oligopoly, as a response to an
increase in their liability; nevertheless, apart of considering the case of product liability, they do not
provide the characterization of the optimum, nor the comparative analysis between di¤erent sharing
arrangements.
Finally, it is also worth quoting the paper by Nussim and Tabbach (2009) who challenge the foundations of the standard model of unilateral accident and care, for the reason that it treats care decisions
as a non-durable input. The model we develop in the rest of the paper considers this as a serious point.
Turning to the analysis of care and output decisions by …rms, we take the view that care expenditures
may be understood as investment in speci…c technology and/or infrastructures (dedicated to safety of
industrial plants) with long run e¤ects (over the period of production). As such, we treat them as is
usual in the IO literature as a productive cost independent of …rms output.
4
3
Model and benchmark
3.1
Setup and assumptions
The situation we are focusing on is one where the good produced provides some bene…ts to society (to
consumers of the good), but accidental events may occur during the production process and victims
in case of accident have no contractual nor market relationship with …rms; in particular, victims and
consumers are not the same persons. Moreover, we are considering a case of multiple liability but in
a sense uncertain, since it is not possible to disentangle the in‡uence of each individual …rm on the
aggregate harm to victims.5
To this end, we introduce a very simple model of imperfect competition, where …rms may harm some
victims/the environment and thus invest in precautionary measures to reduce the cost of liability. We
consider the market for an homogenous product, where two …rms compete à la Cournot. Both consumers
and …rms are risk neutral. The quantity of goods produced by …rm i is denoted qi (i = 1; 2), and
Q = q1 + q2 represents the aggregate output of industry. The market demand is given by P (Q) = a
bQ,
(a > 0; b > 0), consumers being not harmed by the product.
We assume that the expected harm H(X; Q) has the form H(X; Q) = Q2 h(X), with X = x1 + x2
and xi representing the level of care of …rm i = 1; 2; moreover, we assume that for any X > 0, h0 (X) < 0,
and h00 (X) > 0. This speci…c assumption captures the fact that the expected harm is related to the
industry output ( @H
@Q = 2Qh(X) > 0), without any possibility to disentangle the in‡uence of each …rm
@H
( @q
=
1
@H
@q2
=
@H
@Q );
2
moreover the e¤ect is cumulative ( @@QH2 = 2h(X) > 0) meaning that the higher the level
of aggregate activity, the higher the marginal (expected) harm. Similarly, …rms’individual precautionary
@H
measures are supposed to have the same (negative) impact on the expected damage ( @X
=
@H
@x1
=
@H
@x2
=
2
@ H
2 00
Q2 h0 (X) < 0), but returns to scale in the care activity are decreasing ( @X
2 = Q h (X) > 0). Finally,
remark that this speci…c functional form may be understood as a case where the probability of accident
is captured by h(X), assuming h(X) < 1, and the damage in case an accident occurs is scaled by the
(square of) output Q2 . Although this is not the unique interpretation, we will adhere to it throughout
the paper.
Let us turn to the productive costs of …rms. Since we want to capture the situation where care is
considered as a speci…c input in the process of production, with a durable nature, we will assume that
the total cost of production of …rm i is Ci (qi ; xi ) = ci (xi ), 8i = 1; 2, with c0i (xi ) > 0 and c00i (xi ) > 0. To
sum up, this speci…cation implies for the sake of simplicity that total production costs of …rm i reduce
to the cost of care ci (xi ), the durable nature of care being captured by the fact that the cost of care
as a speci…c input of production does not depend on …rm’s activity level (since it is independent from
5 Firms are located at the same place and experience an accident at the same moment; or the damage is di¤use, being
the outcome of several minor failures in the production process which are not observable by outside parties, but having
large and cumulative e¤ects above some threshold on victims or the environment in the long run etc.
5
the cost of the other productive inputs used by a …rm to produce the good); moreover, it is transparent
that we assume that the marginal cost of the other productive inputs is constant and null6 (as generally
considered in the IO literature) - hence with regard to the use of (other) productive inputs, both …rms
are supposed to be identical.
However, we will introduce some heterogeneity between …rms, assuming that at any given level of
precaution, both the total and marginal cost of precaution are lower for …rm 1 than for …rm 2: 8x :
c1 (x) < c2 (x) and c01 (x) < c02 (x).
3.2
The benchmark: social welfare maximization
We …rst determine the socially optimal level of care and output, associated with the maximization of
RQ
social welfare, which is de…ned as the sum of consumers’total utility SC = 0 P (z)dz minus the total
production costs of the output (including the cost of care) augmented of the expected harm. A benevolent
planner can directly use the fact that decisions regarding care activity and output are separated from a
technological point of view, since care is durable (i.e. does not interfere with the use of other productive
inputs). Hence, given that …rms are identical in terms of marginal cost of production and produce a
homogenous good (…rms outputs are perfect substitutes for consumers), the social planner will allocate
the same level of output to each …rm: q1 = q2 = q such that Q = 2q. As a result, social welfare is given
by:
W (q1 ; q2 ; x1 ; x2 ) = aQ
b 2
Q
2
c1 (x1 )
c2 (x2 )
Q2 h(X)
(1)
w
and the …rst-order conditions for an interior solution in Qw ; xw
1 ; x2 are written:
a
bQ =
h0 (X):Q2
2Qh(X)
(2)
= c0i (xi ), 8i = 1; 2
(3)
Condition (2) means that, at the social optimum, increasing the output level must be pushed to the
point where the marginal market proceeds are equal to the marginal cost associated with expected harm.
From conditions (3), we deduce that the social optimum requires, for both …rms, that the marginal cost
of care expenditures equals the marginal bene…t associated with the decrease in liability (expected harm).
Remark that using equations (2) and (3), the optimal output levels (both aggregate and individual) may
be written as a best response function to X w : Qw =
a
b+2h(X w )
= 2q w , which are strictly increasing in X w .
By the same token, the optimal levels of care may also be written as a best response to (decreasing in)
the care level of the other competitor, implicitly de…ned by:
6 In
h0 (X w ):
words, we assume that 8i = 1; 2, Ci (qi ; xi ) = ci (xi ) + kqi , with k = 0.
6
a
b+2h(X w )
2
= c0i (xw
i ), 8i = 1; 2.
0
w
Remark that (3) implies c01 (xw
1 ) = c2 (x2 ) at the optimum, and thus due to the di¤erence in marginal
w
costs of care, we also have : xw
1 > x2 .
4
Equilibrium with liability sharing
Having characterized the optimal solution, we now turn to the analysis of liability. We remind that
we consider here a situation where the individual in‡uence of each …rm on the aggregate expected harm
H(X; Q) cannot be disentangled. For that reason, we consider that Courts set for the liability of both …rms
in solidum, and decide that damages to victims are shared between both …rms. Let us denote as Li (X; Q)
the amount of compensation accruing to …rm i = 1; 2; we will consider alternative arrangements in the
next paragraphs, according to which, …rm’s i liability is calculated such that Li (X; Q) = si
H(X; Q)
where si is …rm’s i liability share (i = 1; 2).
4.1
Care and output under per capita apportionment
Let us assume …rst that strict liability is augmented with a damage rule consisting in an equal share of
the damage between the …rms: they have to compensate victims with an equal share of in expected harm
(s1 = s2 = 12 ).
In this case, …rm i = 1; 2 chooses a level of output and care in order to maximize its pro…t
max
qi ;xi
i (qi ; xi )
= (a
bQ)qi
ci (xi )
1
2
Q2 h(X)
(4)
The …rst-order conditions for …rm i (denoting the other competitor as j) require that :
a
2bqi
bqj
1
h0 (X): Q2
2
= Qh(X)
(5)
= c0i (xi )
(6)
Obviously, (5)-(6) have the same interpretation compared to (2)-(3). However, two characteristic
features are noticeable here. First comparing equation (5) to (2), we observe that …rms do not fully bear
the total harm to victims (RHS in (2), the marginal increase in the excepted harm). Due to the per capita
apportionment, they only support half of it (RHS in (5)). Second, comparing equation (6) and (3), a
similar e¤ect appears, given that each …rm invests in care activity according to their private marginal
bene…t (LHS in (6)) re‡ecting the decrease in expected liability, which is half of the social marginal bene…t
(LHS in (3)). We return to this in more details in the last paragraph, however these simple observation
suggest that strict liability with equal damage sharing rule introduces distortions both on the output and
care levels.
7
Note that turning now to the equilibrium, we verify indeed that equilibrium values of the output and
care levels are di¤erent from the optimal ones: it can be observed that the …rms being identical in terms
of productive marginal cost (the in‡uence of care activity on the cost of the other inputs being null),
condition (5) is identical for both …rms: this implies that …rms at equilibrium produce the same a level
of output, q1 = q2 = q. Using (5) for i; j = 1; 2 we obtain the equilibrium aggregate and individual
output levels which are given by respectively: Qpc =
2
a
3 b+ 23 h(X pc )
= 2q pc where X pc is the aggregate
expenditures in care. Now, using (6), the equilibrium levels of care xpc
i for i = 1; 2 is implicitly de…ned
as a best response to (decreasing in) the aggregate care level by:
h0 (X pc ): 92
a
b+ 23 h(X pc )
2
= c0i (xpc
i ),
pc
8i = 1; 2. Once more, due to the di¤erence in marginal costs of care, we also observe that: xpc
1 > x2 .
4.2
Care and output under market share apportionment
Let us assume now that strict liability is augmented with a damage rule based on the individual market
share of each …rm: …rms have to compensate only a share of the expected harm determined in proportion
to their market share (si =
qi
Q,
i = 1; 2).
Firm i = 1; 2 chooses now a level of output and care which maximize the pro…t :
max
qi ;xi
i (qi ; xi )
= (a
bQ)qi
ci (xi )
qi
Q
Q2 h(X)
(7)
The …rst-order conditions for …rm i (denoting its competitor as j) require that :
a
2bqi
bqj
=
(Q + qi ) h(X)
h0 (X):qi Q = c0i (xi )
(8)
(9)
Once more, (8)-(9) have the same interpretation compared to (2)-(3). However, two characteristic
features are noticeable here. Comparing conditions (8) and (2), it is obvious that once more …rms do not
fully bear the social cost of their market activity (RHS in (2)), and in thus the marginal cost due to their
individual liability is smaller (RHS in (8)) than at optimum. Finally, comparing equation (9) and (3),
we conclude that in setting of their care expenditures, …rms only consider their private marginal bene…t
(LHS in (9)) re‡ecting the decrease in their expected liability, which is smaller than at optimum (LHS
in (3)). Hence, the same conclusion applies here: strict liability with a damage sharing rule based on
market share introduces distortions both on the output and care levels.
We verify here that the equilibrium does not coincide with the optimum. Using (8) which is similar
for i = 1; 2, it can be concluded once again that …rms produce the same level of output, q1 = q2 = q.
Thus the solution to equation (8) correspond to Qms ; q ms respectively the aggregate and individual levels
of output, which are given by: Qms =
2
a
3 b+h(X ms )
= 2q ms , with X ms the aggregate level of care. In
8
turn using (9) the equilibrium levels of care xms
for i = 1; 2 is implicitly de…ned as a best response to
i
(decreasing in) the aggregate care level:
h0 (X ms ): 92
a
b+h(X ms )
2
= c0i (xi ), 8i = 1; 2. Obviously, due to
mc
the di¤erence in marginal costs of care, we also observe that: xmc
1 > x2 .
5
Discussion
5.1
Comparison of per capita vs market share liability rules
We now compare the per capita rule of apportionment with the market share rule of apportionment.
Let us …rst compare conditions (2) and (5), using that at equilibrium under both liability sharing
rules, both …rms produce the same quantity; this implies that (2) and (5) may be equivalently written
as:
a
3bq
=
2qh(X)
a
3bq
=
3qh(X)
In words, they have the same LHS (marginal market proceeds, which decrease in q); but both RHS
(corresponding to the marginal cost of liability) verify all else equal (for a given X) 2qh(X) < 3qh(X). As
a result, it comes that for a given value of aggregate care expenditures, X, we obtain that q pc > q ms (and
thus Qpc > Qms ). The intuition of the result is as follows. As we know from …rst order conditions, a …rm
chooses its level of output such that the marginal market proceeds equal the increase of the individual
expected liability (victims’expected compensation accruing to her). However, under the per capita this
latter increases proportionally to the market output, whereas, under the market share apportionment it
increases more than proportionally to the market output. Since the marginal bene…t is the same in both
situations, then …rms always produce more under a per capita rule than under a market share rule of
apportionment.
However, we have to take into account the feedback e¤ect of care activity levels (indeed, X is not the
same but is speci…c to each liability regimes, as we now explain). For that purpose, let us turn now to
conditions (6) and (9); given that under both liability sharing rules, the levels of output satisfy Q = 2q,
(6)-(9) can also be written the same way as:
h0 (X):2q 2 = c0i (xi ); 8i = 1; 2
In contrast to what holds for the determination of market outputs, they have the same RHS (the
marginal cost of care, which increases in xi ). However the di¤erence in both LHS (corresponding to the
individual marginal bene…t of liability, decreasing in care) re‡ects the in‡uence of the market equilibrium.
9
2
2
Thus since we have seen that (for a given X) q pc > q ms – implying thus 2 (q pc ) > 2 (q ms ) , it comes
ms
8i = 1; 2 (and thus X pc > X ms ).
that xpc
i > xi
To complete the argument, let us return to the determination of the output levels, and assess the
feedback e¤ect of care activity on equilibrium outputs. Note that since X pc > X ms , we obtain h(X pc ) <
h(X ms ): this means the cost of liability passed to each …rm with the adjustment of precautionary measures
is smaller under the …rst regime (equal sharing) than under the second (market share), implying a smaller
operating cost for …rms, and thus reinforcing the di¤erential e¤ect on the level of market output initially
analyzed.
We summarize the results in the next proposition:7
Proposition 1. When liability is allotted between …rms according to their market share, both the aggregate
market supply and the aggregate expenditures in care are smaller at equilibrium than when liability is
equally shared among …rms (Qms < Qpc ; X ms < X pc ; h(X ms ) > h(X pc )). In contrast, the equilibrium
expected damage may be larger as well as smaller (Qms :h(X ms ) ? Qpc :h(X pc )).
It is important to remark that under both liability sharing rules, …rms obtain exactly the same
market share at equilibrium, which is
1
2.
Thus, the impact of liability sharing is mainly driven by the
market adjustment: liability sharing implies a contraction in the individual level of output (and thus the
contraction of the market supply allows an increase in the equilibrium price), this one in turn providing
…rms with incentives to reduce their expenditures in precaution. As previously discussed, this market
adjustment is more important under the liability regime based on market share liability rule, than under
the equal share rule. In a sense, this means that the consumers of the good are more a¤ected (in terms
of surplus/utility loss) under the market share liability rule, than under the equal sharing arrangement.
In contrast, the e¤ect on victims in terms of expected damage is ambiguous, meaning that whether the
market share is better or worse than the equal share liability regime in controlling risky activities, is still
an open issue.
Indeed, from the ex post point of view, victims are better o¤ under the market share liability rule, than
under the equal sharing arrangement: in cases where an accident occurs, the e¤ective damage to victims
is smaller under the market share liability rule, than under the equal sharing arrangement. However, as
a result of weaker incentives to invest in care provided by the market share liability rule, compared to
the equal sharing arrangement, the probability of accident is also larger with the …rst rule. Hence, the
ex ante point of view does not allow to conclude.
that under a no liability regime, the equilibrium corresponds to a level of output equal to Qnl = 2a
> Qpc
3b
n
o
2
2
2a 2
pc
pc
ms
associated with no investment in care, such that the expected harm is 3b h(0) > max (Q ) h(X ); (Q ) h(X ms ) .
7 Remark
10
5.2
On the ine¢ ciency of liability sharing rules
Finally, we consider here the issue of the distortions introduced by strict liability under the di¤erent
liability sharing arrangements considered here.
For that purpose, let us write conditions (2) and (3) as (using once more that at optimum: Q = 2q)
a
2bq
h0 (X):4q 2
=
4qh(X)
= c0i (xi ), 8i = 1; 2
The …rst line shows that compared with Cournot competition (whatever the liability sharing rule), the
optimal level of output is the result of two opposite e¤ects: on the one hand, a higher marginal bene…t
(LHS in the …rst equation) in terms of market proceeds; on the other hand, a higher marginal cost of
liability (respectively RHS of the …rst equation, for a given X). The net e¤ect is thus ambiguous, all else
equal, and the optimal individual output (and thus the optimal market supply) may be smaller, as well
as larger than under any regime of liability sharing.
The second line shows now that, when both the accident externality (full expected damage) and the
market externality are jointly internalized, the marginal bene…t (LHS) associated with care expenditures
is, all else equal (speci…cally for a given output level Q or q), larger than under any regime of liability
sharing - the marginal cost of precautionary measures being the same; thus, all else equal (for a given q)
this suggests the tendency for optimal care expenditures to be larger than the equilibrium one8 . However,
the …nal size of the marginal bene…t of care depends on the output level, meaning that given the ambiguity
in the comparison between q w , q pc or q ms , then the comparison between optimal care expenditures and
equilibrium ones is also ambiguous.
The results are summarized in the last proposition:
Proposition 2. Whatever the damage sharing arrangement adopted (equal share vs market share), strict
liability augmented with a damage sharing rule leads to an ine¢ cient outcome, both in terms of output
and care expenditures, under Cournot competition and durable care. However, both the direction and size
of the distortions is undetermined: the equilibrium levels of output and care expenditures may be too high,
as well as too low compared the optimal ones.
Simple calculations may help in understanding the di¤erent forces driving the total e¤ect. Since the
market adjustment process is all in this framework, we can calculate for example:
8 This
tendancy is exhacerbed (mitigated) when the optimal output is larger (smaller) than at the Cournot equilibrium.
11
qw
qw
q pc
=
a b 4 h(X w ) 21 h(X pc )
6 (b + 2h(X w )) b + 32 h(X pc )
q ms
=
a b 4 h(X w ) 43 h(X ms )
6 (b + 2h(X w )) (b + h(X ms ))
This suggests that, in a case where the market demand is weakly (highly) sensitive to price – in
the sense that b is large (respectively, small) – there is a pressure for the optimal output to be set
at a higher (lower) level compared to the Cournot level, whatever the sharing arrangement prevailing.
However, the feedback in‡uence of the di¤erence in care expenditures and probability of accident, may
work in di¤erent directions. Remark speci…cally that it is not necessary that h(X w ) > h(X ms ) to have
an opposite tendency for the optimal output to be smaller than at equilibrium: if h(X w ) is not too small
compared to h(X ms ), then h(X w )
6
3
ms
)
4 h(X
> 0 and h(X w )
1
pc
2 h(X )
> 0 may hold.
Conclusion
The paper shows that although it is easy to compare the consequences of liability under the di¤erent
regimes of damage sharing for the consumers of the good, things are less clear for the victims. As a main
conclusion, it is not clear that the market share rule is dominating the per capita rule in contexts where
there is a hard uncertainty that prevents from disentangling the in‡uence of each …rm on the occurrence of
accidents and harm borne by victims. Under strict liability with damage sharing, …rms have an incentive
to reduce the output level, because of the additional cost due to liability load, the contraction in output
being larger under the market share rule than under the equal sharing arrangement –despite …rms share
half of the market under both regimes at equilibrium. In turn, as the output decreases, …rms’ liability
exposure is reduced, justifying to cut individual expenditures in care. As a result, regarding the issue of
the control of risk through tort law and liability regime, there is a trade-o¤ between a smaller damage
in case of accident (market share rule) or a smaller probability of accident (equal share arrangement).
Turning to the issue of e¢ ciency, the analysis is quite deceptive. Compared to the discipline imposed by
liability under imperfect competition, the optimal level of output is in‡uenced by contradictory forces
(both the marginal bene…t and the marginal cost associated with production increase), which cast some
doubts on the general structure of incentives exerted on care expenditures.
This work may be extended in several directions, to start with, relaxing the assumption of …rms
symmetry in terms of production cost, and/or turning to the case of non durable care activities. As
the cost structure of …rms changes in both situations, introducing cross e¤ects between care and other
productive inputs, it can be anticipated that the analysis of strategic interactions at the market stage
and incentives to invest in precaution will become richer but less clear-cut. However, it is important to
12
analyze the e¤ect of liability sharing on competition and care expenditures, and mainly how it a¤ects
…rms market shares at equilibrium. A neglected aspect in the arguments of pros and cons the market
share apportionment solution, comes from the fact that these market shares are not exogenous, but they
re‡ect the structure of incentives designed by tort law and liability rules, given the characteristic features
of the competitive environment. Hence, assessing the in‡uence of alternative competitive environments
is also at the top of our agenda of research.
7
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